Study of a homographic function

Statement

Study the homographic function $y = \dfrac{3x-1}{x+2}$ (asymptotes, centre, intersections with the axes).

Solution

With $a=3$, $b=-1$, $c=1$, $d=2$ we have $ad-bc=3\cdot 2-(-1)\cdot 1=7\ne 0$: it is a genuine homographic function.

Vertical asymptote: set the denominator to zero, $x+2=0 \iff \boxed{x=-2}$.

Horizontal asymptote: as $|x|\to\infty$, $y\to\tfrac{3x}{x}=3$, hence $\boxed{y=3}$.

Centre of symmetry: $O'\left(-\tfrac{d}{c};\tfrac{a}{c}\right)=\boxed{O'(-2;3)}$.

Intersections with the axes:

• with $Oy$ ($x=0$): $y=\tfrac{3\cdot 0-1}{0+2}=-\tfrac{1}{2}$. Point $\left(0;-\tfrac{1}{2}\right)$.
• with $Ox$ ($y=0$): $3x-1=0 \iff x=\tfrac{1}{3}$. Point $\left(\tfrac{1}{3};0\right)$.

Links

Topics: Homographic hyperbola
Concepts: Asymptote · Horizontal asymptote · Vertical asymptote · Centre of symmetry · Homographic function
Functions: Homographic function
Methods: Canonical hyperbola
Skills: Analytic geometry · Studying a function
Exercise type: Function study